I am reading a famous book Introduction to the Theory of Linear Partial Differential Equation written by Chazarain and Piriou. In page 63, the authors define $C^\infty(\overline{\Omega})$ is the space of functions $\phi \in C^\infty(\Omega)$ such that $\partial^\alpha\phi \in C^0(\overline{\Omega})$, for any multi-index $\alpha$.
Here $\Omega$ is a regular open subset of $\mathbb{R}^n.$ (For the definition of regular open set, see also page 63. But it is not important here.) The authors then define $C_0^\infty(\overline{\Omega})$ for the functions $\phi \in C^\infty(\overline{\Omega})$ with compact support in $\overline{\Omega}.$
I am confused about the definition of $C^\infty(\overline{\Omega}).$ We only know $\phi \in C^\infty(\Omega),$ how can we define $\partial^\alpha \phi(x)$ for $x \in \partial \Omega$?
Besides, for $\phi \in C_0^\infty(\overline{\Omega}),$ is $\phi(x)$ for $x\in \partial \Omega$ necessarily $0$? We know that $\text{supp}\phi \subset \overline{\Omega}$ and $\text{supp}\phi$ is $K \cap \overline{\Omega},$ where $K$ is a compact set in $\mathbb{R}^n.$ I don't think it can assure $\phi(x)=0$ for $\in \partial \Omega.$ Am I right?
Best Answer
Perhaps they mean that for every $x_0\in\partial\Omega$, the limit $\lim_{x\to x_0}\partial \phi(x)=g(x_0)$ exists, and that the function $g$, which equals $\partial \phi(x)$ in $\Omega$, is continuous in $\overline{\Omega}$.
Usually the space $C^\infty(\overline{\Omega})$ is the space of smooth functions that can be extended to a smooth function on an open set that contains the closure of $\Omega$.
As for the second, you are right. You cannot assume that the function is zero on the boundary. Any constant function belongs to $C^\infty([0,1])$. But if your set is unbounded, then you know that your function is zero near “infinity”. In particular, if $\Omega$ is the entire space, then your functions are zero outside a compact set.